Eight Versions of Bayes’s Theorem
Name
Formu la
Com ments
This is the simplest form, derivable from the
conjunction rule. When H Ö E, it reduces to
P(H|E) = P(H) / P(E), an important special
case
Simple form
P(H|E) =
P(H) P(E|H)
P(E)
Explicit form
P(H|E) =
P(H) P(E|H)
P(H) P(E|H) + P(~H) P(E|~H)
This is obtained from the simple form by
expanding P(E) according to the theorem on
total probability (TTP)
General form
P(H 1|E) =
P(H1) P (E |H 1)
P(H 1) P (E |H 1) + ... + P(H n) P E|H n)
This is again obtained from the simple form
by the TTP on the assumptio n that H 1, ..., H n
form a partition
Sigma form
P(H 1|E) =
Canceled form
P(H|E) =
P(H 1) P (E |H 1)
Gi P(H i) P (E |H i)
1
1+
Odd s form
Relative odds
form
Compound odds
form
P(~H)
P(E|~H)
×
P(H)
P(E|H)
P(H|E)
P(H)
=
P(~H|E)
P(~H)
P(H 1|E)
=
P(H 2|E)
P(H 1)
P(H 2)
This is a compact way of writing the general
form, of which it is a notational variant
This is obtained from the explicit form by
dividing the numerator and the denominator
by P(H)P(E|H) and collecting terms. It is
very useful when we replace E with the
testimony of multiple independent witnesses
×
P(E|H)
P(E|~H)
This is useful because, as in the canceled
form, the term P(E) has dropp ed out. The
likelihood ratio on the right is the “Bayes
factor,” the term by which the prior odds are
multiplied to obtain the posterior odds
×
P(E |H 1)
P(E |H 2)
This generalization of the o dds form is
useful when we do not have a partition but
we wish to know the relative impact of E on
two hyp otheses.
P(H|A&B )
P(H)
P(A|H)
P(B|H)
=
×
×
P(~H|A&B)
P(~H)
P(A|~H)
P(B|~H)
This form allows combination of evidence,
assuming that ±H screens off A from B.
W ithout this assumption, the probabilities in
the final term would have to be conditional
on A as well. Given this assumption, we can
combine evidence by multiplying the prio r
odds by successive B ayes factors.
Copyright © Tim McGrew 2005